Prince William, Duke of Cambridge and Catherine, Duchess of Cambridge and Prince Harry join Team Heads Together at a London Marathon Training Day - Credit: Chris Jackson /Getty

The development at the Grade 1-listed Orangery would provide more than 1,500 sq m of extra space over three floors - two below ground, and one above ground.

The planning application submitted by HRP to Kensington and Chelsea Council last week says: “The addition of a basement storey is required to allow for the accommodation of administration which must necessarily be moved out of rooms leased from the Royal Household in Kensington Palace.”

Two-storey basement extensions are usually in breach of the strict new policy on basements introduced by Kensington & Chelsea Council to combat the spread of ‘iceberg’ developments in the borough.

The orangery - Credit: John Warburton-Lee/Alamy

But planning officials will be urged to approve the plans on the grounds that an “exception” to the rule that no basement should be more than one floor can be made in the case of “large sites”.

Historic Royal Palaces also argues that the proposed basement will take up less than 50 per cent of the “wider garden area of the site”.

The Duke and Duchess of Cambridge will have 22 rooms and two kitchens in their refurbished apartment at Kensington Palace, or KP as William’s mother Diana, Princess of Wales used to call it.

Princess Eugenie and her boyfriend, Jack Brooksbank, are moving in to Ivy Cottage in the grounds of the palace later this year and Prince Harry lives in nearby Nottingham Cottage, where he has been visited by his girlfriend Meghan Markle.

Kensington and Chelsea council said it could not comment on the plans ahead of them being discussed at by officials later this year.